car won't start! help!

in the past two weeks my 1994 corolla has not been starting sometimes. last time this happened (last week), I was able to get someone to jump-start it and it drove for about a week. and now today, when I went to start it, it wouldn't. it seems like the extremely cold/snowy weather we having been having lately is what makes it not start (I live in CT)- last week's failure to start was after a -1 degree day, and today it's been snowing all day/night.
I think the battery is probably the only problem- my trunk door doesn't shut ever since I was rear-ended a year ago, and so I think having it open (it's tied shut with wire) all the time sapped the battery (having it open means the trunk lights are always on). any other ideas as to what would be causing this?

If the lights are on 24/7, then that's what's killing your battery...that and the cold. If you disconnect the trunk light, your battery will last a lot longer.